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Is simplicity still possible for a more accurate approximation to the perimeter of the ellipse? or, Using the exponential function to further improve the second Ramanujan's approximation

Salvador E. Ayala-Raggi, Manuel Rendón-Marín

TL;DR

This work tackles the long-standing problem of approximating the ellipse perimeter with a closed-form expression by starting from Ramanujan II and modeling the residual error as a decaying exponential in the eccentricity parameter. It introduces two corrections, $P1$ and $P2$, that respectively use one and two exponential terms to adjust Ramanujan II into highly accurate, compact single-line formulas. Fitted via minimax criteria on structured domains, these corrections achieve a maximum relative error around 0.57 ppm across the full range of axis ratios, outperforming Cantrell, Koshy, Moscato, and other compact formulas while maintaining simplicity. The key contribution is a four-parameter, exponentially corrected closed-form that provides exceptional accuracy with minimal computational overhead, making it among the simplest yet most precise ellipse-perimeter approximations available.

Abstract

The perimeter of an ellipse has no exact closed-form expression in terms of elementary functions, and numerous approximations have been proposed since the eighteenth century. Classical formulas by Fagnano, Euler, and Ramanujan, as well as modern refinements such as Cantrell and Koshy methods, aim to reduce the approximation error while maintaining computational simplicity. In this paper, we introduce a new closed-form expression that enhances Ramanujan second formula by dividing it by 1 minus a binomial of two exponential terms resulting in a very stable approximation in a range of b/a between 1 and 1/10000, or even up to a smaller ratio. The resulting approximation remains compact, requiring only four constants, and achieving a remarkable tradeoff between simplicity and accuracy. Across the full eccentricity range of b/a in [0.0001,1], our method attains a maximum relative error of approximately 0.57 ppm with respect to the exact perimeter computed via elliptic integral. Our formula is quasi-exact at the extremes, for the circle b/a=1 and for the degenerate flat ellipse b/a=0. Compared with Cantrell approximation, the proposed method reduces the maximum relative error by a factor of 25 while preserving a short and elegant expression. This makes it one of the simplest yet most accurate closed-form and single-line approximations to the ellipse perimeter currently available in the literature.

Is simplicity still possible for a more accurate approximation to the perimeter of the ellipse? or, Using the exponential function to further improve the second Ramanujan's approximation

TL;DR

This work tackles the long-standing problem of approximating the ellipse perimeter with a closed-form expression by starting from Ramanujan II and modeling the residual error as a decaying exponential in the eccentricity parameter. It introduces two corrections, and , that respectively use one and two exponential terms to adjust Ramanujan II into highly accurate, compact single-line formulas. Fitted via minimax criteria on structured domains, these corrections achieve a maximum relative error around 0.57 ppm across the full range of axis ratios, outperforming Cantrell, Koshy, Moscato, and other compact formulas while maintaining simplicity. The key contribution is a four-parameter, exponentially corrected closed-form that provides exceptional accuracy with minimal computational overhead, making it among the simplest yet most precise ellipse-perimeter approximations available.

Abstract

The perimeter of an ellipse has no exact closed-form expression in terms of elementary functions, and numerous approximations have been proposed since the eighteenth century. Classical formulas by Fagnano, Euler, and Ramanujan, as well as modern refinements such as Cantrell and Koshy methods, aim to reduce the approximation error while maintaining computational simplicity. In this paper, we introduce a new closed-form expression that enhances Ramanujan second formula by dividing it by 1 minus a binomial of two exponential terms resulting in a very stable approximation in a range of b/a between 1 and 1/10000, or even up to a smaller ratio. The resulting approximation remains compact, requiring only four constants, and achieving a remarkable tradeoff between simplicity and accuracy. Across the full eccentricity range of b/a in [0.0001,1], our method attains a maximum relative error of approximately 0.57 ppm with respect to the exact perimeter computed via elliptic integral. Our formula is quasi-exact at the extremes, for the circle b/a=1 and for the degenerate flat ellipse b/a=0. Compared with Cantrell approximation, the proposed method reduces the maximum relative error by a factor of 25 while preserving a short and elegant expression. This makes it one of the simplest yet most accurate closed-form and single-line approximations to the ellipse perimeter currently available in the literature.
Paper Structure (10 sections, 16 equations, 2 figures, 1 table)

This paper contains 10 sections, 16 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Materials and Methods
  3. One–exponential correction to R2 ($R2/1exp$)
  4. Two–exponential correction ($R2/2exp$)
  5. Experimental Results
  6. Computational Setup
  7. Relative Error Comparison (Selected Methods)
  8. Summary of Maximum Errors
  9. Discussion
  10. Key insight

Figures (2)

  • Figure 1: a) Signed relative error comparison of ellipse perimeter approximations. The graph shows that $R2/1exp$ (purple) outperforms the others methods when a<100, while $R2/2exp$ (blue) outperforms all methods across the entire range ($a<10000$). b) Absolute relative error comparison and three references of elliptical orbits of famous comets.
  • Figure 2: Closed-form ellipse perimeter approximations.